First excited state of a 100*100 box vs number of iterations (N) - - PowerPoint PPT Presentation

First excited state of a 100*100 box vs number of iterations (N) Starting from a random state Graphing:

Second excited state of a 100*100 box vs number of iterations (N) Starting from a random state Graphing:

The first excited state should be doubly-degenerate Ø Lanczos only gives one state out of a degenerate multiplet Go back to the Krylov space If states k, j are degenerate, we have a term For any m, this vector points in the same direction in the subspace spanned by Acting with H cannot “separate” degenerate states Since the Lanczos basis spans the same Krylov space, we

• nly get one state out of a degenerate multiplet of states

Ø the particular linear combination depends on the initial state Numerical round-off errors can lead to apparent degeneracies (multiple copies of the same state). This indicates that the scheme breaks down as the basis becomes non-orthogonal.

Example in two dimensions: box with open boundaries

labeling for 4*4 elements subroutine hoperation(f1,f2) f2(:)=vpot(:)*f1(:) do j=1,nx*ny x=1+mod(j-1,nx) y=1+(j-1)/nx if (x.ne.1) f2(j-1)=f2(j-1)-t*f1(j) if (x.ne.nx) f2(j+1)=f2(j+1)-t*f1(j) if (y.ne.1) f2(j-nx)=f2(j-nx)-t*f1(j) if (y.ne.ny) f2(j+nx)=f2(j+nx)-t*f1(j) enddo Constructing State n stored in f1(nx*ny) State constructed in f2(nx*ny) t = hopping (kinetic) matrix element (open corresponds to hard walls)

• consider hopping into all boxes j

One step in the iteration of the a and b coefficients

if (m==1) then call hoperation(f0,f1) aa(0)=dot_product(f0,f1) f1=f1-aa(0)*f0 nn(1)=dot_product(f1,f1) else call hoperation(f1,f2) aa(m-1)=dot_product(f1,f2)/nn(m-1) bb(m-2)=nn(m-1)/nn(m-2) f2=f2-aa(m-1)*f1-bb(m-2)*f0 nn(m)=dot_product(f2,f2) f0=f1 f1=f2 endif

The method of constructing the normalized states directly is very similar (program on-line) here m=n+1

The full basis and Hamiltonian construction

f0(:)=psi(:) nn(0)=1.d0 Do m=1,niter perform code on previous page enddo d(:)=aa(:) e(:)=sqrt(bb(:)) call diatri(niter,d,e,eig,states)

Random initial state

do i=1,n psi(i)=rand()-0.5d0 enddo norm=1.d0/sqrt(dot_product(psi,psi)) psi(:)=psi(:)*norm

Perform niter

niter Lanczos steps and diagonalize

Calculation of the states

In order to calculate states (wave functions) we have to perform another Lanczos procedure, since we have not saved all the states |fn> If we want the m-th lowest state, we transform with the m-th eigenvector obtained in the diagonalization. The eigenvectors are in the matrix states; vec=states(:,m

vec=states(:,m) f0=psi f0=psi psi=psi*vec(0) psi=psi*vec(0) call hoperation(n,f0,f1) call hoperation(n,f0,f1) f1=f1 f1=f1-aa(0)*f0 aa(0)*f0 psi=psi+vec(1)*f1/sqrt(nn(1)) psi=psi+vec(1)*f1/sqrt(nn(1)) do i=2,niter do i=2,niter-1 call hoperation(n,f1,f2) call hoperation(n,f1,f2) f2=f2 f2=f2-aa(i aa(i-1)*f1 1)*f1-bb(i bb(i-2)*f0 2)*f0 psi=psi+vec(i)*f2/sqrt(nn(i)) psi=psi+vec(i)*f2/sqrt(nn(i)) f0=f1 f0=f1 f1=f2 f1=f2 enddo enddo Normalized states